The Lie algebra of polynomial vector fields on the affine space with constant divergence is $1.5$-generated

Abstract

A Lie algebra is said to be $1.5$-generated if every nonzero element can be completed to a two-element generating set. We prove that the Lie algebra of polynomial vector fields with constant divergence on the affine space is $1.5$-generated. We also show that the Lie algebra of all polynomial vector fields on the complex affine space is generated by two completely integrable elements.